UvA-DARE (Digital Academic Repository)
Initiating planet formation : the collisional evolution of small dust aggregates
Paszun, D.
Link to publication
Citation for published version (APA):
Paszun, D. (2008). Initiating planet formation : the collisional evolution of small dust aggregates
General rights
It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),
other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating
your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask
the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,
The Netherlands. You will be contacted as soon as possible.
UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl)
Download date: 17 Jun 2017
1
Introduction
For centuries people have been looking at the sky and watching a slow and
graceful play of the stars and the planets. These mysterious dots have been attracting attention of scientists and philosophers. They tried to understand the
structure of the Universe and our place in it. Thanks to their efforts, during the
past several hundred years, we have gained a great understanding of our world
and the Universe as a whole. Now we know that the Solar System is not special
(we have observed over 200 other planetary systems) and that the Earth is orbiting the Sun, unlike what was believed before Copernicus (1543) published his
famous “De revolutionibus orbium coelestium”. Yet still so much remains unclear.
We still do not know, how the Earth or any other known planet (out of over
3001 ) was formed. Below we present the current state of our understanding of
planet formation mechanism. We also introduce terminology that is commonly
used in the literature and further in this thesis.
1.1 Star formation
A commonly accepted scenario of planet formation starts with a collapse of a
molecular cloud core2 . Due to the conservation of angular momentum, the rotation velocity of the contracting nebula increases. The gravitational force is
partially opposed by the centrifugal force and the material falling in forms an
active accretion disk. During this stage the protostar is completely obscured by
the nebula and can be observed only in the far infrared and mm wavelengths.
1 The source - http://exoplanet.eu - gives 307 planets outside the Solar System as for July
25th 2008.
2 For a detailed description of this stage of star formation we refer the reader to studies of Ebert
(1955); Bonnor (1956); Shu (1977), and Shu et al. (1987).
1
2
Introduction
Such objects are categorized as class 0 and schematically look as in Fig. 1.1a.
Figure 1.1 — Four stages of star and planet formation. The associated timescales are
indicated below each panel.
Already during the initial collapse of the molecular cloud dust grains collide
and form larger complexes. Suttner and Yorke (2001) studied this growth in the
collapse of the molecular cloud cores and found that dust particles can grow to
several tens of micrometers. This limit depends on the assumed composition of
dust grains and in particular their adhesion properties (for further discussion
see Sect. 1.5.1 and Sect. 6).
As the mass accretion onto the central star proceeds, a bipolar stellar wind
may initiate and start clearing out the falling in cloud. Such objects are still
embedded in the parental cloud and are characterized by the disk dominated
spectra. They are classified as class I sources (see Fig. 1.1b). The material in the
surrounding nebula is eventually exhausted and the remaining nebula is cleared
by the stellar wind. This results in a dramatic decrease of the mass accretion rate.
The disk enters a passive phase and the star becomes visible for the first time in
visible and UV wavelengths. This characterizes class II objects (see Fig. 1.1c).
At this stage, dust particles grow many orders of magnitude in mass forming
planets. After approximately 10 million years the gas is dispersed from the disk
or is accreted onto giant gaseous planets. As the dust is mostly incorporated
into the largest bodies, the signature of the remaining material (debris disk) can
be observed at long wavelengths (class III sources).
1.2 The geometry of disks
3
1.2 The geometry of disks
Dust particles suspended in the cloud fall onto a newly formed disk, where they
can continue to coagulate to larger structures - aggregates (also referred to later
as agglomerates or particles). This growth is driven by relative velocities that originate due to several processes. A schematic picture of a disk is given in Fig. 1.2.
Figure 1.2 — Sketch of a protoplanetary disk with indicated different processes.
Grains can settle to the midplane collecting other particles. Moreover turbulence stirs the gas and dust is dragged along. Therefore particles that respond
to the gas friction fast will collide with smaller, well coupled aggregates. Finally
the particles can move radially towards the star, as the friction of the gas takes
the angular momentum from dust particles. The growth of dust to larger sizes
is the first and very important step towards the formation of planets. The collisional accumulation must proceed to at least centimeter to meter sizes, before
any other process can take over. Therefore understanding of this first step is
crucial to obtain a complete picture of the formation of the Solar System.
The growth of dust also plays an important role in shaping the disk structure.
The central star irradiates the surface of the disk. The absorption is mainly due
to small dust particles that forward the energy to the gas. Higher temperature
of the gas results in puffing up. The disk height increases with radius resulting
in a flaring shape. These flaring disks are called group 1 and are characterized
by a relatively strong IR excess in the spectrum. The dust grows in the disk and
can significantly change this picture. Agglomerates settle to the midplane when
pulled in by gravity. This removes the opacity source and the disk becomes
transparent to the radiation. As a result the gas stops being heated and falls
4
Introduction
down. The inner regions, however, are still heated and remain puffed-up. They
provide a shadow that blocks the radiation from reaching the outer disk. These
sources we call group 2 and they are characterized by a weaker IR excess in their
spectra.
The growth of dust can be observed indirectly. It is indicated by the slope in
the mm range of the observed spectral energy distribution or in the shape of a
10 μm silicate feature. Many disks show evidence of the process of dust growth
taking place (van Boekel et al. 2003; Acke et al. 2004; Bouwman et al. 2008). This
growth in longer perspective leads to the formation of planets.
1.3
Models of planet formation
The formation of planets is still not fully understood and two main routes of
research are followed in parallel exploring different possible scenarios. The first
one is the gravitational instability of the disk (e.g., Boss 1997) that fragments into
self-gravitating clumps of gas and dust that upon further contraction form giant
gaseous planets on short timescales (∼ 103 years). The second competing scenario is the core accretion model (Pollack 1984), where the gas is accreted onto
a rocky core that forms first. This scenario involves much longer time scales
(∼ 107 years), as the accretion of gas must be preceded by the collisional accumulation of dust particles in large bodies ( of about ∼ 10 M⊕ ). Therefore dust
grains have to grow many orders of magnitude in mass, before the gas can even
start to accrete onto the protoplanetary core. Here we present the two scenarios
with their strong and weak points.
1.3.1
Gravitational instability of the disk
The gaseous disk intermixed with solids can develop an instability and form
gravitationally bound clumps. In this scenario, giant, gaseous planets form in
one quick step. The disk, however, must be massive enough and cool on short,
orbital, time scales, for the clumps to form (Boss 2000). If such clumps can form
and survive, the dust grains sediment to their centers and form the rocky core
of a giant planet.
A number of models have been employed to study gravitational instability
of the disk with different results that strongly depend on assumed conditions
(Durisen et al. 2007). Therefore this model of giant planet formation is still hotly
debated (Pickett et al. 2000; Boley et al. 2006, 2007; Mayer et al. 2002, 2004, 2007).
1.3.2
Gravitational collapse of the dust midplane
Cores of giant planets can also form in two steps scenario, initially proposed
by Goldreich and Ward (1973). They assumed that in a laminar disk dust can
settle to a thin midplane that can reach a critical thickness and become gravitationally unstable. This leads to the fast formation of large, about km-sized
1.4 The formation of planetesimals
5
bodies – planetesimals. Then gravity speeds up the growth to massive cores (it
increases the collisional cross-section of planetesimals). However, for this mechanism to work, the disk must be extremely laminar. In reality shear between
dust and gas layers will induce a Kelvin-Helmholtz instability that prevents the
dust midplane from reaching the critical thickness (Weidenschilling 1984; Cuzzi
et al. 1993; Johansen et al. 2006a).
1.3.3
Core accretion model
The core accretion scenario of planet formation (Pollack 1984; Pollack et al.
1996) consists of two stages. In the first one dust particles collide and stick to
each other forming larger agglomerates. This collisional accumulation of solids
speeds up, once km-sized bodies are formed. Gravity then becomes important
and enhances the collision cross-section of massive boulders leading to runaway
growth (Lissauer 1987). This results in the formation of large solid cores with
masses of about 10 M⊕ . During this growth, they collect a gaseous envelope that
eventually accretes onto the core of this protoplanet. The timescale of the formation of a giant planet in the core accretion model is limited by the gas dissipation
from the protoplanetary disk (∼ 107 years).
The collisional accumulation of material (the first stage of the core accretion
model) is also the necessary mechanism in forming rocky, Earth-like planets.
Before any other mechanism can take over, particles must grow to at least cm or
meter sizes. In the following sections we focus our attention on the first steps
that lead to the formation of planetesimals – about km-sized bodies that are not
affected anymore by the interaction with gas.
1.4 The formation of planetesimals
The formation of planetesimals requires the dust particles to grow about 27 orders of magnitude in mass (from ∼ μm size grains to km size planetesimals)
before their gravity becomes important. Here we outline the growth phases
starting with the initial coagulation of the smallest building blocks - monomers.
1.4.1
The Brownian growth phase
Microscopic dust grains, respond to gas motion very fast. They are continuously
hit by gas molecules from different directions, which influences their trajectories. These random impacts are reflected in the motion of the particles3 . As each
grain moves in a random direction, dust particles can collide with each other
and stick due to the attractive van der Waals force (see Sect. 1.5.1 for discussion
on the microphysics of contact between dust grains). The growth in these conditions is quasi monodisperse, meaning that at each given time the size distribu3 This
random motion of particles was first observed by Brown (1828).
6
Introduction
tion is very narrow. Therefore, collisions occur preferentially between particles
of similar mass. As a consequence aggregates obtain fractal structure (Krause
and Blum 2004, chapter 2), meaning that the packing density profile is given by
the power law:
R Df
N(R) = kf
.
(1.1)
r0
Here N(R) is the number of particles of radius r0 inside a sphere with radius
R around the center of mass of the aggregate, Df is the fractal dimension, and
kf is the fractal prefactor. Depending on the mean free path of a dust particle,
aggregates of different fractal dimension can form. Low gas densities present
far away from the central star, result in Brownian motion that is characterized
by a mean free path of dust particles longer than their size (ballistic collisions)
and cause the formation of particles with fractal dimension of about Df ≈ 1.5.
Higher gas densities result in a decrease of the mean free path and also a decrease of the fractal dimension (Paszun and Dominik 2006). In the inner part
of the protoplanetary disk, at high gas density, extremely elongated aggregates
(Df ≈ 1) can form.
As particles grow to larger sizes, they begin to decouple from the gas, which
terminates the Brownian growth phase, as the relative velocities start to be determined by other mechanisms.
1.4.2
The compaction phase
When particles start to decouple from the gas, their relative velocities increase
(see Sect. 1.4.4). Larger particles that respond to the gas on a longer time scale,
move with respect to the gas that is followed by still tightly coupled, small aggregates. Therefore aggregates of very different size will collide preferentially,
leading to the formation of particles with homogeneous, non fractal structure
(Df = 3.0) (Ball and Witten 1984). This, however, still results in very porous
aggregates. Thus when dust grains acquire high enough relative velocities restructuring can occur. This process can dissipate part of the collision energy
and permit further growth even for velocities that normally result in breakup
of aggregates. The compression phase, however, cannot last indefinitely. Blum
et al. (2006) have shown that aggregates can be compressed to limited densities,
where the volume fraction of the solids is about 33%. The compression of aggregates increases their density, until further restructuring becomes very difficult.
Upon further compression aggregates expand but conserve the final density of
about 33%.
1.4.3
The erosion and fragmentation phase
The compression of aggregates terminates eventually. The relative velocities
increase as an effect of decoupling of dust particles due to both an increase of
their mass and an increase of compactness. Therefore the ratio of the mass over
1.4 The formation of planetesimals
7
the projected surface increases resulting in an increase of the response time of
dust on the gas drag (see Sect. 1.4.4).
As grains are held together by van der Waals forces, there is a threshold of
energy which will break these bonds. An increase of the relative velocities inevitably leads to high energy, destructive collisions. These have been studied
theoretically (Dominik and Tielens 1997) and in laboratory experiments (e.g.,
Blum and Muench 1993; Blum and Wurm 2000; Wurm et al. 2005b; Langkowski
et al. 2008a). Dominik and Tielens (1997) have shown that the sticking ability
depends very much on the properties of monomers. Small monomers are characterized by higher threshold for sticking than larger particles. Also chemistry
has a great influence. Ice particles can survive much higher collision energy than
silicate aggregates. Thus, depending on conditions, the onset of the fragmentation can be shifted to lower or higher velocities. However, once it is reached,
particles can be shattered and further growth is in jeopardy.
1.4.4
Sources of the relative velocity
The relative velocities between dust aggregates can be generated by different
mechanisms. For small particles, well coupled to the gas, the Brownian motion
dominates. For larger particles that start to decouple from the gas other sources
of the relative velocity are important. The quantity that actually determines
these velocities is the stopping (or friction) time. For grains in the Epstein regime
(when the grain size is smaller than the mean free path of a gas molecule) this
response time to the gas drag is given by
τf =
3 m
,
4 ρ cs σd
(1.2)
where ρ is the gas density, m is the mass of the dust particle, and σd is its projected surface area. The isothermal sound speed in the gas cs is given by
kB T
cs =
,
(1.3)
μ mp
where μ is the mean molecular weight, kB is the Boltzmann’s constant, T is the
gas temperature, and mp is the proton mass. Particles with a different ratio of
their mass over the projected surface m/σd respond to the gas drag on different timescales τf and develop relative motion, the magnitude of which depends
obviously on the difference of the stopping times of the particles. The relative
speed of dust aggregates can originate from different mechanisms. Below we
present a few that are the most important during the formation of planetesimals.
• The Brownian motion
Particles tightly coupled to the gas move collectively and only very small
relative velocities are present. This relative velocity between two particles
8
Introduction
due to the Brownian motion is given by
Δv =
8kB T
,
πmμ
(1.4)
where mμ = m1 m2 /(m1 + m2 ) is the reduced mass of the two particles with
masses mi . For small, micron-sized grains in a protoplanetary disk, this
velocity is of the order of a few mm/s and leads to perfect sticking.
• Vertical settling
The vertical component of the gravitational force pulls aggregates in the
direction of the midplane. If particles are very well coupled to the gas, the
turbulence can keep them high above the midplane, because the settling
timescale is much longer than the turbulent mixing. The settling velocity
for particles is given by
(1.5)
vsett = Ω2 z τf ,
where Ω is the Kepler orbital frequency and z is the height above the
midplane. A dust particle that settles with this velocity will collide with
smaller grains that are descending slower (due to a lower value of τf ).
• Turbulence
Beside the normal orbital motion, the gas undergoes turbulence. Since
dust particles respond to the gas drag with some delay τf , they can cross
trajectories of other aggregates that already coupled to the gas or the ones
that have even longer friction time and move independently of the gas.
Therefore particles can acquire very high relative velocities of up to several
times 10 m/s, depending on the strength of turbulence (Voelk et al. 1980;
Weidenschilling and Cuzzi 1993; Ormel and Cuzzi 2007).
The turbulent motion is commonly described in terms of different scale
fluctuations that transfer the energy from the largest eddies down to the
smallest scales, where this energy is dissipated. The smallest scale relates
to the largest scale through the Reynolds number Re as τs = Re−1/2 τL ,
where τs and τL denote the overturn times of the smallest and the largest
eddies, respectively. Similarly the velocity and the length scale as vs =
Re−1/4 vL and ls = Re−4/3 lL , respectively. The Reynolds number is defined
by the turbulent viscosity νT that was introduced to explain the angular
momentum transfer in the disk, as the molecular viscosity νm was too low.
This turbulent viscosity is given by
νT = αcs /Hg ,
(1.6)
where Hg = cs /Ω is the pressure scale height and α is a dimensionless
constant, commonly used to parameterize the strength of the turbulence
(Shakura and Sunyaev 1973).
1.4 The formation of planetesimals
9
• Radial drift
Dust particles in a protoplanetary disk orbit the central star with the Kepler velocity
G M
,
(1.7)
vK = Ω r =
r
where G is the gravitational constant and r is the distance from the star
with mass M . The gas in the disk on the other hand feels an additional
force due to the radial pressure gradient (1/ρg dP/dr) and thus moves on a
sub-Keplerian velocity
(1.8)
vg = Ωg r,
where Ωg is given by
Ω2g r =
G M
1 dP
.
+
ρg dr
r2
(1.9)
This results in a relative velocity between the gas and dust particles, that
in the case of negative pressure gradient (a decreasing pressure with radial distance) causes a head wind for the dust grains slowing them down.
They transfer their angular momentum to the gas and spiral inwards at
substantial radial velocities. In the case of meter sized bodies this inward
velocity is of the order of 104 cm/s (Weidenschilling 1977). Note that if for
some reason the pressure gradient is locally positive (pressure increases
with radial distance), dust particles feel back wind that will move them
away from the star. Such local, high pressure regions will accumulate dust
particles and can stimulate growth.
1.4.5
Limitations
As presented in Sect. 1.4.3 and Sect. 1.4.4, the formation of planetesimals has to
face several obstacles. Here these problems are presented along with comments
on possible solutions and work that has been done in relation to these issues.
• Collisional fragmentation
Very small dust particles are well coupled to the gas and their relative velocities are small, as they all move together. Thus collisions lead to sticking and growth to larger sizes. However, as coagulation progresses, particles respond to the gas motion on longer timescales. This results in an
increase of the relative velocities. Therefore the progressing dust coagulation causes an increase of the collision energy and leads ultimately to the
disruption of particles at high relative velocities.
Dominik and Tielens (1997) have shown that dust aggregates can restructure during a collision causing a dissipation of a fraction of the kinetic energy. Depending on the composition of dust particles, this mechanism can
10
Introduction
be more efficient, leading to an increase of the destruction energy threshold or less efficient resulting in fragmentation at slower impacts. Therefore, it is expected that very porous particles that can undergo substantial
restructuring, can grow to larger masses, as the collision energy can be
more efficiently dissipated.
Recently, Brauer et al. (2008b) have studied the growth of dust particles
in a local density maximum due to the evaporation front at the snow line.
This high pressure region of the disk can accumulate dust particles. In the
midplane of the disk, weaker turbulence (Ciesla 2007) results in a lower
relative velocities. Brauer et al. (2008b) have shown that for the fragmentation threshold velocity of at least 5 m/s, the growth can proceed to large
boulders of up to several 100 meters in size. However, a more realistic value of the threshold velocity, for aggregates made of micron-sized
grains, is ∼ 1 m/s (Blum and Muench 1993; Poppe et al. 2000; Langkowski
et al. 2008a). Therefore, it is crucial, to investigate further the structure of
growing particles and the effect it has on the collisional outcome.
From the experimental side Wurm et al. (2005b) have shown that collisions of large, mm to cm-sized, precompacted aggregates can lead to a net
growth at velocities far beyond the fragmentation threshold.
• Meter size barrier
Weidenschilling (1977) has shown that the head-wind, due to subKeplerian orbital velocity of gas component, has the strongest effect for
approximately meter-sized particles. The resulting radial drift velocity is
of the order of 104 cm/s, meaning that at one astronomical unit4 they can
move all the way to the star in just about 100 years. In fact, the dust is
not falling into the star but rather is destroyed in the innermost region of
the disk, where high temperatures (∼ 1500 K) evaporate solids. This can
efficiently prevent the growth beyond meter-sized boulders.
Recently, Johansen et al. (2007) have shown that large meter-sized boulders can accumulate in high pressure turbulent eddies forming dense
clumps. These undergo further, order of magnitude, increase in density
due to streaming instability driven by a relative motion of dust and gas,
resulting in gravitationally bound clusters. Thus the meter size barrier can
be crossed, as the collapse of these over-dense clumps occurs on timescales
shorter than in the case of the radial drift. Even in this case, however, it
is absolutely necessary to grow to larger centimeter or meter-sizes first,
before this mechanism can take over.
Structure and composition of dust particles determines under what collision
conditions growth or destruction takes place. The outcome of a collision, therefore, not only depends on energy but also on aggregate structure. Dust growth
4 One astronomical unit (1 AU) is the mean distance between the Sun and the Earth and is equal
to 1.496 · 1013 cm.
1.5 General concepts
11
Figure 1.3 — Scheme of a contact deformation. Two grains of radii R1 and R2 in contact
form the contact area of radius a. The displacement δ is indicated.
models must take this into account. Aim of this thesis is to provide the physical
description of collisional growth including effects of grain morphology.
1.5 General concepts
1.5.1
Microphysics of dust particles
Dust coagulation is possible because small grains feel an attractive force that
can hold them bound together. The nature of this force can be diverse. A few
examples are: dipole-dipole interaction for ices and electrostatic and magnetic
forces for charged and magnetized particles. The lower limit for the attractive
force is the very short-range van der Waals force.
Spherical grains in contact pushed towards each other with some force Fex
deform at their surfaces.
This forms a flat, circular contact area that stores an
elastic energy Eel = Fex dδ. This energy depends on the material properties
and deformation of the surface. The displacement δ is presented in Fig. 1.3, a
schematic picture of two spherical grains in contact. Realistically, the real force
acting on the two grains is the sum of the external applied force and the attractive surface force F = Fex + Fs . This leads to the formation of a contact area with
a radius a given by (Johnson et al. 1971)
a=
3R 1/3
2 + 12πγRF
+
6πγR
±
F
,
(6πγR)
ex
ex
4E ∗
(1.10)
where R = (1/R1 + 1/R2 )−1 is the reduced radius, γ is the surface energy and E ∗
12
Introduction
Figure 1.4 — Degrees of freedom of a contact between two grains (after Dominik and
Tielens 1997): a - stretching, b - rolling, c - sliding, d - twisting.
is the reduced elasticity modulus given by
E∗ =
1 − ν2
1
E1
+
1 − ν22 −1
.
E2
(1.11)
Symbols Ei and νi denote the Young’s moduli of two grains and their Poisson
ratios, respectively. The centers of two grains are separated by the distance R1 +
R2 − δ (see Fig. 1.3), where δ is the displacement due to the exerted force and is
given by
a2 a 2 4 a 1/2 −
,
(1.12)
δ= 0 2
2R a0
3 a0
with a0 given in Eq. (1.13a).
In the absence of the external force Fex , the contact radius and the displacement δ are in equilibrium (the elastic repulsion is balanced by the attractive
force)
9πγR2 1/3
,
(1.13a)
a0 =
E∗
δ0 =
a20
.
3R
(1.13b)
A system of two grains in contact has 6 degrees of freedom5 as presented in
Fig. 1.4. They are:
Stretching
The vertical degree of freedom is presented in Fig. 1.4a. When grains are pulled
away from each other, the contact area decreases and, when the displacement δ
5 Note that rolling and sliding have 2 degrees of freedom each. For a complete discussion of this
subject we refer to an excellent paper by Dominik and Tielens (1997).
1.5 General concepts
13
approaches zero, a neck of material (as greatly exaggerated in Fig. 1.4a) is pulled
of the grains (Chokshi et al. 1993; Dominik and Tielens 1997) resulting in the
negative displacement δ. To separate the two grains one needs to overcome the
attractive potential Eel , which leads to the definition of the pull-off force
Fc = 3πγR
(1.14)
and the critical displacement
1 a20
.
(1.15)
2 61/3 R
Chokshi et al. (1993) have shown that grain collisions lead to excitation of elastic
waves and partial dissipation of the energy. Moreover when a contact is broken,
additional waves are excited, meaning that the energy necessary for the two
particles to bounce is higher than the elastic energy stored in the contact Eel .
The required energy is actually given by (Dominik and Tielens 1997)
δc =
Ebr = 1.8Fc δc = 43
γ5/3 R4/3
.
E ∗2/3
(1.16)
This translates directly to the breaking velocity
vbr = 1.07
γ5/6
E ∗1/3 R5/6 ρ1/2
d
,
(1.17)
where ρd is the bulk density of the dust grain. This relation shows that an
increasing grain size results in a decrease of the breaking velocity. Therefore,
larger grains bounce easier and small monomers stick at larger velocities. The
breaking velocity for micron-sized silica grains (r0 = 0.6 μm, γ = 25 erg/s,
E = 5.4 · 1011 dyn/cm2 , ν = 0.17, ρd = 2.65 g/cm3 ) is about 30 cm/s.
Experiments by Poppe et al. (2000) showed, however, that this velocity is underestimated, as monomers were sticking to a flat surface at velocities of about
120 cm/s. Therefore Eq. (1.16) has to be scaled to correspond to experimental
results. This has been done by Paszun and Dominik (2008b), who used a plastic deformation of small surface asperities as an additional energy dissipation
mechanism.
Rolling
The rolling degree of freedom is presented in Fig. 1.4b. As the grains roll over
each other, the contact area must follow. Therefore, in the direction of motion
a new contact area forms, while behind, the contact breaks. This is caused by
asymmetric distribution of pressure that is positive (compressed contact) in the
front and negative (stretched contact) behind. This pressure distribution leads
to a torque that opposes the rolling motion (Dominik and Tielens 1995)
M = 4Fc
a 3/2
ξ,
a0
(1.18)
14
Introduction
where ξ is a linear distance that the contact area is lagging behind. The contact actually shifts and irreversible motion occurs, when the grains roll a critical
distance ξcrit . The energy needed to roll a contact over this distance is given by
2
.
eroll = 6πγξcrit
(1.19)
This, however, corresponds to, typically, inter-atomic distances of about 2Å (Dominik and Tielens 1995). Therefore it is good to define a rolling energy that corresponds to a visible rearrangement, e.g., angular distance of 90◦ ( corresponding to a linear distance of πR for two equal size spherical monomers). Thus, the
rolling energy required for a visible restructuring is given by
Eroll = 6π2 γRξcrit .
(1.20)
Also in this case experiments have been performed to verify the theory.
Heim et al. (1999) have shown that the rolling energy is actually larger. The
reason is that the critical distance ξcrit is in fact about 10 times larger - of the
order of 20Å.
Sliding
The sliding degree of freedom is presented in Fig. 1.4c. Contrary to the rolling
degree of freedom, irreversible sliding can occur only if almost the entire contact is broken (Dominik and Tielens 1996), as the contact area has to shift entirely. Therefore, the energy needed to begin sliding motion is comparable to
the breaking energy and is given by
eslide =
1
F2 ,
16aG∗ fric
(1.21)
where G∗ = ((2 − ν1 )/G1 + (2 − ν2 )/G2 )−1 and Gi is the shear modulus of i-th grain.
The friction force Ffric is given by
Ffric =
G 2
πa + Fatomic ,
2π
(1.22)
where G = (1/G1 +1/G2 )−1 is the reduced shear modulus and Fatomic is the friction
force due to dissipation of the energy on atomic scales (Dominik and Tielens
1997). By analogy to rolling the sliding motion is irreversible when the contact
shifts by a critical distance δcx . The visible restructuring occurs then at energies
associated with sliding over the distance πR. Thus the energy is simply
Eslide = eslide
πR 1
= πRFfric .
δcx
2
(1.23)
1.5 General concepts
15
Twisting
In twisting (see Fig. 1.4d) the friction force is determined by the same physical
mechanism as the sliding motion. Grains rotate around the axis normal to the
contact area. Therefore the surfaces slide over each other. In this case the energy
associated with irreversible motion is
etwist =
3
2
(Mcrit
twist ) ,
32Ga3
(1.24)
Ga3
+ Matomic ,
3π
(1.25)
where the torque Mcrit
twist is given by
Mcrit
twist =
with Matomic being the torque due to dissipation of energy on atomic scales (Dominik and Tielens 1997). Here the energy associated with the visible restructuring is the energy needed to twist over an angle of π/2. Therefore the energy is
simply
π
Etwist = Mcrit
.
(1.26)
2 twist
1.5.2
Structure of aggregates
The growth of dust aggregates results in structural changes, that can significantly affect the coagulation, i.e., the stopping time τf of the particles can change.
The structure of dust aggregates can be described using various quantities. Here
we present a few of the most important structural parameters, that are widely
used in the literature.
The fractal dimension
The fractal dimension Df (see Eq. (1.1)) describes the scaling of the mass of an
object with size. Therefore the possible range of Df is narrow and is limited from
the bottom by Df ≥ 1 for the most fluffy particles - linear strings. The upper limit
of the fractal dimension is Df ≤ 3, and corresponds to homogeneous particles
with a constant density structure throughout nearly the entire aggregate.
Although linear strings are uniquely defined by the fractal dimension alone,
more complex particles with higher values of Df can have many possible forms.
The reason is that the fractal dimension does not define the density, but the
density structure. Therefore the central density must be provided for a complete
description. In Eq. (1.1) the index kf fulfills this requirement.
The volume filling factor
Although the fractal dimension provides a complete description of the structure, more intuitive quantities are widely used. A classic example is the volume
16
Introduction
filling factor φ. It is defined as the ratio of the volume occupied by the solids to
the total volume, i.e.,
Vcompact
φ=
.
(1.27)
Vtot
For aggregates of size Rout , made of finite number N of spherical monomers of
single size r0 , this quantity is given by
φ=N
r 3
0
,
Rout
(1.28)
and is limited by the densest packing of spheres - close cubic packing (CCP)6 .
This gives the upper limit of φCCP = 3 √π 2 ≈ 0.74048. This high filling factor
requires a high level of organization and is very unlikely to occur through a
random process7 . The densest random packing on the other hand is about φ ≈
0.635 (Onoda and Liniger 1990).
The lower limit of the filling factor depends on the size of an aggregate and
approaches zero for infinitely large particles. The lowest filling factor is obtained
for a linear string of N monomers and equals φ = N −2 . This limit for aggregates
of the arbitrary fractal dimension Df is given as
φ = kf3/Df N 1−3/Df .
(1.29)
The geometrical filling factor
The geometrical filling factor φσ is conceptually similar to the volume filling
factor. However, it has one significant advantage: it provides information about
the projected surface area σd and is given by
φσ =
Vcompact
.
4
3/2
3 π(σd /π)
(1.30)
For aggregates made of a finite number of monomers N of a fixed size r0 , this is
given by
r 3
0
φσ = N
,
(1.31)
Rσ
√
with the projected surface equivalent radius Rσ = σd /π. Therefore, the friction
time is related to the geometrical filling factor as
1/3
.
τf ∝ φ2/3
σ N
6 The
(1.32)
same density can be obtained via two different arrangements of monomer layers - face
centered cubic packing and hexagonal close packing.
7 Structures with spheres packed so densely can form in natural environments. An example is the
precious opal - a rock made of densely packed silica spheres that cause interference and diffraction
of transmitted light resulting in different colors (Klein and Hurlbut 1985).
1.6 This thesis
17
Both filling factors (φσ and φ) are basically equal for compact, non-fractal aggregates. The projected surface equivalent radius Rσ is in this case nearly equal
to the physical radius Rout . Fluffy, fractal particles, however, differ significantly,
as the physical size can be much larger than the equivalent of the projected surface.
1.5.3
Coagulation from numerical perspective
The growth of dust is conventionally followed in terms of the mass distribution f (m)dm. The distribution function f (m)dm provides the number of particles
with masses in the mass interval between m and m + dm. As particles grow,
the shape and position of the mass distribution can evolve depending on the
type of growth (e.g., the Brownian growth phase is characterized by an almost
monodisperse distribution, meaning that the initial distribution will only shift
towards larger sizes while its shape remains the same).
For a very large dynamical range (distribution spans many orders of magnitude in mass), however, it is useful to present the distribution for logarithmic
mass intervals dlog m as
f (m) dm = ln10 f (m) m dlog m.
(1.33)
The evolution of the mass distribution is commonly performed using the
Smoluchowski coagulation equation (Smoluchowski 1916)
d f (m)
1
= − f (m) K(m, m ) f (m )dm +
K(m , m − m ) f (m − m ) f (m )dm . (1.34)
dm
2
The first term in Eq. (1.34) indicates the particles of mass m that are removed,
as they coagulate with other particles. The second term is the source of particles of mass m that form due to coagulation of smaller particles, the masses of
which add up to m. Note the factor of 12 that prevents counting particles twice.
The function K(m, m ) in Eq. (1.34) is the coagulation kernel and provides the
probability of collision between particles of mass m and m .
Another method to describe evolving particle systems, is the stochastic
Monte Carlo method. Instead of integrating Eq. (1.34), it “collides” individual
particles. This approach, however, is limited by the resolution of dynamical
scales. Processes that last over the shortest timescales must be resolved, meaning that the longest timescales may not be well represented. This issue, however,
was addressed by Ormel and Spaans (2008) and successfully solved by grouping of small particles, which significantly speeds up simulation at the smallest
scales.
1.6 This thesis
In this thesis we provide a quantitative description of some of the most important processes that are directly involved in the coagulation of dust aggregates. In
18
Introduction
chapters 2 and 3 we present the development of porous structures in the Brownian and hierarchical growth phases in the framework of the hit-and-stick energy
regime. This gives rise to initially fractal aggregates, that later in the hierarchical growth contribute to the formation of very porous, non-fractal particles.
Chapter 2 shows that previously neglected rotation of particles can significantly
affect the structure of aggregates formed in the Brownian growth phase. Chapter 3 shows a significance of hierarchical growth that was believed not to play
an important role.
In chapter 4 we present a state of the art model of three-dimensional microscopic aggregates. It takes into account surface forces that hold monomers
together and are responsible for aggregates dynamics. This new tool allows us
to understand microphysics of dust growth and simulate compaction and fragmentation of porous aggregates. We test this model extensively against results
of laboratory experiments. Our model is further used to determine material
properties of a porous medium, showing a new area for future applications.
The sound speed in a porous medium is determined along with the compression curve.
Chapter 5 presents results of an extensive parameter study of collisions of
small dust aggregates. The influence of the porosity, the impact parameter, and
energy is described in detail that has not been known before. These results are
used to provide a quantitative recipe of the collision outcome that now truly
opens the possibility to study both the growth of dust aggregates and their structural evolution during the coagulation.
The final chapter 6 provides results of dust coagulation in molecular cloud
cores. The collision recipe is formulated to fit the needs of the Monte Carlo
model. This provides a very powerful, new method that accounts for not only
the mass but also the structure of aggregates. We also test how sensitive our
method is to modifications in the recipe. We compare the default model that
includes the full recipe with several cases: the coagulation of only compact particles (structural parameter is kept constant throughout the growth), and coagulation assuming central collisions only. This shows the importance of structural
effects on the growth of dust particles.